to constructing model category structures in the category of all unbounded chain ... Hovey [19]; in particular, a generalization of the main results of [15] to classes ...
KAPLANSKY CLASSES, FINITE CHARACTER, AND ℵ1 –PROJECTIVITY ˇ JAN SAROCH AND JAN TRLIFAJ Abstract. Kaplansky classes emerged in the context of Enochs’ solution of the Flat Cover Conjecture. Their connection to abstract model theory goes back to [4]: a class C of roots of Ext is a Kaplansky class closed under direct limits, iff the pair (C, ≤) is an abstract elementary class (AEC) in the sense of Shelah. We prove that this AEC has finite character in case C = ⊥ C ′ for a class C ′ of pure–injective modules. In particular, all AECs of roots of Ext over any right noetherian right hereditary ring R have finite character (but the case of general rings remains open). If (C, ≤) is an AEC of roots of Ext then C is known to be a covering class. However, Kaplansky classes need not even be precovering in general: We prove that the class D of all ℵ1 –projective modules (which is equal to the class of all flat Mittag–Leffler modules by [18]) is a Kaplansky class for any ring R, but it fails to be precovering in case R is not right perfect, the class ⊥ (D⊥ ) equals the class of all flat modules and consists of modules of projective dimension ≤ 1. Assuming the Singular Cardinal Hypothesis, we prove that D is not precovering for each countable non–right perfect ring R.
Introduction A class A of (right R–) modules is a Kaplansky class provided that there is an infinite cardinal κ such that for each 0 6= A ∈ A and X ⊆ A with |X| ≤ κ, there exists a ≤ κ–presented module A′ ∈ A such that X ⊆ A′ ⊆ A and A/A′ ∈ A. Kaplansky classes naturally occur in algebra, homotopy theory, and model theory. The fact that the class FL of all flat modules over an arbitrary ring is a Kaplansky class was crucial for proving the Flat Cover Conjecture in [6]. In [12] it was shown that Kaplansky classes are important sources of module approximations. In [15] the notion was extended to Grothendieck categories G, and applied to constructing model category structures in the category of all unbounded chain complexes over G. In all these cases the focus was on Kaplansky classes closed under direct limits. In parallel, deconstructible classes of modules have widely been used as set– theoretic tools of homological algebra in [8], [9], [10], [16] et al. (see Definition 1.3 below). Recently it has been shown in [13] that deconstructible classes provide an appropriate setting for application of the homotopy–theoretic methods of Hovey [19]; in particular, a generalization of the main results of [15] to classes not necessarily closed under direct limits was obtained.
Date: November 14, 2010. 2000 Mathematics Subject Classification. 16D80, 03C95, 16D40, 03E35. Key words and phrases. Kaplansky class, Ext, deconstructible class, abstract elementary class, finite character, ℵ1 –projective module, Singular Cardinal Hypothesis, precover of a module. ˇ 201/09/H012, the second by GACR ˇ 201/09/0816 and MSM First author supported by GACR 0021620839. Both authors supported by the PPP program MEB 101005. 1
ˇ JAN SAROCH AND JAN TRLIFAJ
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There is a close relation between Kaplansky classes and deconstructible classes: Let C be a class of modules closed under transfinite extensions. If C is deconstructible then C is a Kaplansky class, and the converse holds when C is closed under direct limits (cf. Lemma 1.5). Many classes C closed under transfinite extensions are actually classes of roots of Ext, that is, they are of the form C = ⊥ C ′ for a class of modules C ′ , where \ ⊥ ′ C = KerExtiR (−, C ′ ) = {M | ExtiR (M, C ′ ) = 0 for all C ′ ∈ C ′ and i ≥ 1}. 1≤i
